Senate Democratic leader Charles Schumer warned that destroying any tapes would break the law.
Republican Senator Lindsey Graham said the White House needed to "clear the air" about whether tapes existed.
The comments come after Mr Trump tweeted what appeared to be a thinly veiled threat to the former FBI chief.
He warned Mr Comey last week against talking to the media, saying he had "better hope there are no tapes" of their conversations.
The White House has neither confirmed nor denied the presence of any tapes.
Mr Schumer also warned that Senate Democrats might refuse to vote on a new FBI director until a special prosecutor is named to investigate alleged Russian meddling in the US election.
The FBI is investigating the allegations and possible ties between Moscow and the Trump campaign.
Mr Trump denies any such links and says Mr Comey had assured him he was not being investigated. He says he fired Mr Comey because he was not doing a good job.
Democrats, however, have accused President Trump of firing Mr Comey to try to thwart the FBI inquiry.